Copper crystallises in fcc with a unit cell length of 361 pm. What is the radius of copper atom?

Options

A. 157 pm
B. 181 pm
C. 127 pm
D. 108 pm

Explanation

To find the radius of a copper atom in a face-centered cubic (FCC) unit cell, we can follow these steps:

### Step 1: Understand the FCC Structure
In a face-centered cubic (FCC) unit cell, atoms are located at each of the corners and the centers of each face of the cube. The relationship between the edge length (A) of the unit cell and the atomic radius (R) can be derived from the geometry of the cube.

### Step 2: Use the Formula for FCC
For an FCC unit cell, the relationship between the edge length (A) and the atomic radius (R) is given by the formula:
\[ R = \frac{\sqrt{2}}{4} A \]

### Step 3: Substitute the Given Value
We are given that the edge length (A) is 361 pm (picometers). We can substitute this value into the formula:
\[ R = \frac{\sqrt{2}}{4} \times 361 \, \text{pm} \]

### Step 4: Calculate the Value of R
Next, we need to calculate the value:
1. First, calculate \(\sqrt{2}\):
\[ \sqrt{2} \approx 1.414 \]
2. Now substitute this value into the equation:
\[ R = \frac{1.414}{4} \times 361 \, \text{pm} \]
3. Calculate \(\frac{1.414}{4}\):
\[ \frac{1.414}{4} \approx 0.3535 \]
4. Now multiply this by 361 pm:
\[ R \approx 0.3535 \times 361 \, \text{pm} \approx 127 \, \text{pm} \]

### Step 5: Conclusion
Thus, the radius of the copper atom is approximately 127 pm.

### Final Answer:
The radius of the copper atom is **127 pm**.

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